Scaling limits of looperased random walks and uniform spanning trees
 Oded Schramm
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Abstract
The uniform spanning tree (UST) and the looperased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of these subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane.
The scaling limits of these processes are conjectured to be conformally invariant in dimension 2. We make a precise statement of the conformal invariance conjecture for the LERW, and show that this conjecture implies an explicit construction of the scaling limit, as follows. Consider the Löwner differential equation
 [Aiz] M. Aizenman,Continuum limits for critical percolation and other stochastic geometric models, Preprint. http://xxx.lanl.gov/abs/mathph/9806004.
 [ABNW] M. Aizenman, A. Burchard, C. M. Newman and D. B. Wilson,Scaling limits for minimal and random spanning trees in two dimensions, Preprint. http://xxx.lanl.gov/abs/math/9809145.
 [ADA] M. Aizenman, B. Duplantier and A. Aharony,Path crossing exponents and the external perimeter in 2D percolation, Preprint. http://xxx.lanl.gov/abs/condmat/9901018.
 [Ald90] D. J. Aldous,The random walk construction of uniform spanning trees and uniform labelled trees, SIAM Journal on Discrete Mathematics3 (1990), 450–465. CrossRef
 [Ben] I. Benjamini,Large scale degrees and the number of spanning clusters for the uniform spanning tree, inPerplexing Probability Problems: Papers in Honor of Harry Kesten (M. Bramson and R. Durrett, eds.), Boston, Birkhäuser, to appear.
 [BLPS98] I. Benjamini, R. Lyons, Y. Peres and O. Schramm,Uniform spanning forests, Preprint. http://www.wisdom.weizmann.ac.il/≈schramm/papers/usf/.
 [BJPP97] C. J. Bishop, P. W. Jones, R. Pemantle and Y. Peres,The dimension of the Brownian frontier is greater than 1, Journal of Functional Analysis143 (1997), 309–336. CrossRef
 [Bow] B. H. Bowditch,Treelike structures arising from continua and convergence groups, Memoirs of the American Mathematical Society, to appear.
 [Bro89] A. Broder,Generating random spanning trees, in30th Annual Symposium on Foundations of Computer Science, IEEE, Research Triangle Park, NC, 1989, pp. 442–447. CrossRef
 [BP93] R. Burton and R. Pemantle,Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transferimpedances, The Annals of Probability21 (1993), 1329–1371. CrossRef
 [Car92] J. L. Cardy,Critical percolation in finite geometries, Journal of Physics A25 (1992), L201L206. CrossRef
 [DD88] B. Duplantier and F. David,Exact partition functions and correlation functions of multiple Hamiltonian walks on the Manhattan lattice, Journal of Statistical Physics51 (1988), 327–434. CrossRef
 [Dur83] P. L. Duren,Univalent Functions, SpringerVerlag, New York, 1983.
 [Dur84] R. Durrett,Brownian Motion and Martingales in Analysis, Wadsworth International Group, Belmont, California, 1984.
 [Dur91] R. Durrett,Probability, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991.
 [EK86] S. N. Ethier and T. G. Kurtz,Markov Processes, Wiley, New York, 1986.
 [Gri89] G. Grimmett,Percolation, SpringerVerlag, New York, 1989.
 [Häg95] O. Häggström,Randomcluster measures and uniform spanning trees, Stochastic Processes and their Applications59 (1995), 267–275. CrossRef
 [Itô61] K. Itô,Lectures on Stochastic Processes, Notes by K. M. Rao, Tata Institute of Fundamental Research, Bombay, 1961.
 [Jan12] Janiszewski, Journal de l'Ecole Polytechnique16 (1912), 76–170.
 [Ken98a] R. Kenyon,Conformal invariance of domino tiling, Preprint. http://topo.math.upsud.fr/≈kenyon/confinv.ps.Z.
 [Ken98b] R. Kenyon,The asymptotic determinant of the discrete laplacian, Preprint. http://topo.math.upsud.fr/≈kenyon/asymp.ps.Z.
 [Ken99] R. Kenyon,Longrange properties of spanning trees, Preprint.
 [Ken] R. Kenyon, in preparation.
 [Kes87] H. Kesten,Hitting probabilities of random walks on ℤ _{ d }, Stochastic Processes and their Applications25 (1987), 165–184. CrossRef
 [Kuf47] P. P. Kufarev,A remark on integrals of Löwner's equation, Doklady Akademii Nauk SSSR (N.S.)57 (1947), 655–656.
 [LPSA94] R. Langlands, P. Pouliot and Y. SaintAubin,Conformal invariance in twodimensional percolation, Bulletin of the American Mathematical Society (N.S.)30 (1994), 1–61.
 [Law93] G. F. Lawler,A discrete analogue of a theorem of Makarov, Combinatorics, Probability and Computing2 (1993), 181–199. CrossRef
 [Law] G. F. Lawler,Looperased random walk, inPerplexing Probability Problems: Papers in Honor of Harry Kesten (M. Bramson and R. Durrett, eds.), Boston, Birkhäuser, to appear.
 [Löw23] K. Löwner,Untersuchungen über schlichte konforme abbildungen des einheitskreises, I, Mathematische Annalen89 (1923), 103–121. CrossRef
 [Lyo98] R. Lyons, A bird'seye view of uniform spanning trees and forests, inMicrosurveys in Discrete Probability (Princeton, NJ, 1997), American Mathematical Society, Providence, RI, 1998, pp. 135–162.
 [MR] D. E. Marshall and S. Rohde, in preparation.
 [MMOT92] J. C. Mayer, L. K. Mohler, L. G. Oversteegen and E. D. Tymchatyn,Characterization of separable metric ℝtrees, Proceedings of the American Mathematical Society115 (1992), 257–264. CrossRef
 [MO90] J. C. Mayer and L. G. Oversteegen,A topological characterization of ℝtrees, Transactions of the American Mathematical Society320 (1990), 395–415. CrossRef
 [New92] M. H. A. Newman,Elements of the Topology of Plane Sets of Points, second edition, Dover, New York, 1992.
 [Pem91] R. Pemantle,Choosing a spanning tree for the integer lattice uniformly, The Annals of Probability19 (1991), 1559–1574. CrossRef
 [Pom66] C. Pommerenke,On the Loewner differential equation, The Michigan Mathematical Journal13 (1966), 435–443. CrossRef
 [Rus78] L. Russo,A note on percolation, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete43 (1978), 39–48. CrossRef
 [SD87] H. Saleur and B. Duplantier,Exact determination of the percolation hull exponent in two dimensions, Physical Review Letters58 (1987), 2325–2328. CrossRef
 [Sch] O. Schramm, in preparation.
 [Sla94] G. Slade,Selfavoiding walks, The Mathematical Intelligencer16 (1994), 29–35.
 [SW78] P. D. Seymour and D. J. A. Welsh,Percolation probabilities on the square lattice, inAdvances in Graph Theory (Cambridge Combinatorial Conference, Trinity College, Cambridge, 1977), Annals of Discrete Mathematics3 (1978), 227–245.
 [TW98] B. Tóth and W. Werner,The true selfrepelling motion, Probability Theory and Related Fields111 (1998), 375–452. CrossRef
 [Wil96] D. B. Wilson,Generating random spanning trees more quickly than the cover time, inProceedings of the Twentyeighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), ACM, New York, 1996, pp. 296–303. CrossRef
 Title
 Scaling limits of looperased random walks and uniform spanning trees
 Journal

Israel Journal of Mathematics
Volume 118, Issue 1 , pp 221288
 Cover Date
 20001201
 DOI
 10.1007/BF02803524
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 Oded Schramm ^{(1)}
 Author Affiliations

 1. Department of Mathematics, The Weizmann Institute of Science, 76100, Rehovot, Israel